Journal of the
Korean Mathematical Society

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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  • 2022-05-01

    The exceptional set of one prime square and five prime cubes

    Yuhui Liu

    Abstract : For a natural number $n$, let $R(n)$ denote the number of representations of $n$ as the sum of one square and five cubes of primes. In this paper, it is proved that the anticipated asymptotic formula for $R(n)$ fails for at most $O(N^{\frac{4}{9} + \varepsilon})$ positive integers not exceeding $N$.

  • 2022-01-01

    Gorenstein sequences of high socle degrees

    Jung Pil Park, Yong-Su Shin

    Abstract : In [4], the authors showed that if an $h$-vector $(h_0,h_1,\dots,h_e)$ with $h_1=4e-4$ and $h_i\le h_1$ is a Gorenstein sequence, then $h_1=h_i$ for every $1\le i\le e-1$ and $e\ge 6$. In this paper, we show that if an $h$-vector $(h_0,h_1,\dots,h_e)$ with $h_1=4e-4$, $h_2=4e-3$, and $h_i\le h_2$ is a Gorenstein sequence, then $h_2=h_i$ for every $2\le i\le e-2$ and $e\ge 7$. We also propose an open question that if  an $h$-vector $(h_0,h_1,\dots,h_e)$ with $h_1=4e-4$, $4e-3<h_2\le {(h_1)_{(1)}}| ^{+1}_{+1}$, and $h_2\le h_i$ is a Gorenstein sequence, then $h_2=h_i$ for every $2\le i\le e-2$ and $e\ge 6$.

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  • 2022-03-01

    Ring isomorphisms between closed strings via homological mirror symmetry

    Sangwook Lee

    Abstract : We investigate how closed string mirror symmetry is related to homological mirror symmetry, under the presence of an explicit geometric mirror functor.

  • 2022-01-01

    The interior gradient estimate for a class of mixed Hessian curvature equations

    Jundong Zhou

    Abstract : In this paper, we are concerned with a class of mixed Hessian curvature equations with non-degeneration. By using the maximum principle and constructing an auxiliary function, we obtain the interior gradient estimate of $(k-1)$-admissible solutions.

  • 2022-03-01

    Goldbach-Linnik type problems with unequal powers of primes

    Li Zhu

    Abstract : It is proved that every sufficiently large even integer can be represented as a sum of two squares of primes, two cubes of primes, two fourth powers of primes and 17 powers of 2.

  • 2022-03-01

    Generalized Killing structure Jacobi operator for real hypersurfaces in complex hyperbolic two-plane Grassmannians

    Hyunjin Lee, Young Jin Suh, Changhwa Woo

    Abstract : In this paper, first we introduce a new notion of generalized Killing structure Jacobi operator for a real hypersurface $M$ in complex hyperbolic two-plane Grassmannians $S U_{2, m} / S\left(U_{2} \cdot U_{m}\right)$. Next we prove that there does not exist a Hopf real hypersurface in complex hyperbolic two-plane Grassmannians $S U_{2, m} / S\left(U_{2} \cdot U_{m}\right)$ with generalized Killing structure Jacobi operator.

  • 2022-05-01

    Zeros of new Bergman kernels

    Noureddine Ghiloufi, Safa Snoun

    Abstract : In this paper we determine explicitly the kernels $\mathbb K_{\alpha,\beta}$ associated with new Bergman spaces $\mathcal A_{\alpha,\beta}^2(\mathbb D)$ considered recently by the first author and M. Zaway. Then we study the distribution of the zeros of these kernels essentially when $\alpha\in\mathbb N$ where the zeros are given by the zeros of a real polynomial $Q_{\alpha,\beta}$. Some numerical results are given throughout the paper.

  • 2022-01-01

    Grothendieck group for sequences

    Xuan Yu

    Abstract : For any category with a distinguished collection of sequences, such as $n$-exangulated category, category of N-complexes and category of precomplexes, we consider its Grothendieck group and similar results of Bergh-Thaule for $n$-angulated categories [1] are proven. A classification result of dense complete subcategories is given and we give a formal definition of K-groups for these categories following Grayson's algebraic approach of K-theory for exact categories [4].

  • 2022-03-01

    Cohomology of torsion and completion of $N$-complexes

    Pengju Ma, Xiaoyan Yang

    Abstract : We introduce the notions of Koszul $N$-complex, $\check{\mathrm{C}}$ech $N$-complex and telescope $N$-complex, explicit derived torsion and derived completion functors in the derived category $\mathbf{D}_N(R)$ of $N$-complexes using the $\check{\mathrm{C}}$ech $N$-complex and the telescope $N$-complex. Moreover, we give an equivalence between the categories of cohomologically $\mathfrak{a}$-torsion $N$-complexes and cohomologically $\mathfrak{a}$-adic complete $N$-complexes, and prove that over a commutative Noetherian ring, via Koszul cohomology, via RHom cohomology (resp. $\otimes$ cohomology) and via local cohomology (resp. derived completion), all yield the same invariant.

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  • 2022-03-01

    A deep learning algorithm for optimal investment strategies under Merton's framework

    Daeyung Gim, Hyungbin Park

    Abstract : This paper treats Merton's classical portfolio optimization problem for a market participant who invests in safe assets and risky assets to maximize the expected utility. When the state process is a $d$-dimensional Markov diffusion, this problem is transformed into a problem of solving a Hamilton--Jacobi--Bellman (HJB) equation. The main purpose of this paper is to solve this HJB equation by a deep learning algorithm: the deep Galerkin method, first suggested by J. Sirignano and K. Spiliopoulos. We then apply the algorithm to get the solution to the HJB equation and compare with the result from the finite difference method.

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May, 2022
Vol.59 No.3

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